B. Addition and Subtraction of Matrices
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Section 7.2 The Algebra of Matrices
EXAMPLE 4
ᮣ
653
Using Technology for Matrix Operations
Use a calculator to compute the difference A Ϫ B for the matrices given.
2
11
A ϭ £ 0.9
0
Solution
ᮣ
Ϫ0.5
3
4
6
6
5
Ϫ4 §
5
Ϫ12
Bϭ £
Ϫ7
10
1
6
11
25
0
Ϫ4
Ϫ5
9
0.75
Ϫ5 §
Ϫ5
12
The entries for matrix A are shown in Figure 7.9. After entering matrix B, exit to
the home screen [ 2nd MODE (QUIT)], call up matrix A, press the ؊ (subtract)
key, then call up matrix B and press . The calculator quickly finds the difference
and displays the results shown in Figure 7.10. The last line on the screen shows the
result can be stored for future use in a new matrix C by pressing the STO key,
calling up matrix C, and pressing .
ENTER
ENTER
Figure 7.9
Figure 7.10
Now try Exercises 21 through 24
ᮣ
Figure 7.11
In Figure 7.10 the dots to the right on the calculator screen indicate there are additional digits or matrix columns that can’t fit on the display, as often happens with larger
matrices or decimal numbers. Sometimes, converting entries to fraction form will provide a display that’s easier to read. Here, this is done by calling up the matrix C, and
using the MATH 1: ᮣ Frac option. After pressing , all entries are converted to fractions in simplest form (where possible), as in Figure 7.11. The third column can be
viewed by pressing the right arrow.
Since the addition of two matrices is defined as the sum of corresponding entries,
we find the properties of matrix addition closely resemble those of real number addition. Similar to standard algebraic properties, ϪA represents the product Ϫ1 # A and
any subtraction can be rewritten as an algebraic sum: A Ϫ B ϭ A ϩ 1ϪB2. As noted in
the properties box, for any matrix A, the sum A ϩ 1ϪA2 will yield the zero matrix Z,
a matrix of like size whose entries are all zeroes. Also note that matrix ϪA is the
additive inverse for A, while Z is the additive identity.
ENTER
Properties of Matrix Addition
Given matrices A, B, C, and Z are m ϫ n matrices, with Z the zero matrix. Then,
B. You’ve just seen how
we can add and subtract
matrices
I. A ϩ B ϭ B ϩ A
¡
II. 1A ϩ B2 ϩ C ϭ A ϩ 1B ϩ C2 S
III. A ϩ Z ϭ Z ϩ A ϭ A
¡
IV. A ϩ 1ϪA2 ϭ 1ϪA2 ϩ A ϭ Z ¡
matrix addition is commutative
matrix addition is associative
Z is the additive identity
ϪA is the additive inverse of A
C. Matrices and Multiplication
The algebraic terms 2a and ab have counterparts in matrix algebra. The product 2A
represents a constant times a matrix and is called scalar multiplication. The product
AB represents the product of two matrices.
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CHAPTER 7 Matrices and Matrix Applications
Scalar Multiplication
Scalar multiplication is defined by taking the product of the constant with each entry
in the matrix, forming a new matrix of like size. In symbols, for any real number k and
matrix A, kA ϭ 3kaij 4.
EXAMPLE 5
ᮣ
Computing Operations on Matrices
4
3
3
1 § and B ϭ £ 0
Given A ϭ £
0 Ϫ3
Ϫ4
1
1
a. 2 B
b. Ϫ4A Ϫ 2 B
1
2
Solution
ᮣ
Ϫ2
6 § , compute the following:
0.4
3
Ϫ2
1
1
a. B ϭ a b £ 0
6 §
2
2
Ϫ4 0.4
3
1 12 2132
1 12 2 1Ϫ22
Ϫ1
2
1
1
ϭ £ 1 2 2102
1 2 2 162 § ϭ £ 0
3 §
Ϫ2 0.2
1 12 21Ϫ42 1 12 210.42
1
1
b. Ϫ4A Ϫ B ϭ Ϫ4A ϩ aϪ b B rewrite using algebraic addition
2
2
1Ϫ12 2 1Ϫ22
1Ϫ42142
1Ϫ42132
1Ϫ12 2132
ϭ £ 1Ϫ42 1 12 2
1Ϫ42112 § ϩ £ 1Ϫ12 2102
1Ϫ12 2162 §
1
1Ϫ42102 1Ϫ421Ϫ32
1Ϫ2 21Ϫ42 1Ϫ12 210.42
Ϫ16 Ϫ12
1
Ϫ32
ϭ £ Ϫ2
Ϫ4 § ϩ £ 0
Ϫ3 § simplify
0
12
2
Ϫ0.2
Ϫ16 ϩ 1Ϫ32 2
Ϫ12 ϩ 1
Ϫ11
Ϫ35
2
ϭ £ Ϫ2 ϩ 0
Ϫ4 ϩ 1Ϫ32 § ϭ £ Ϫ2
Ϫ7 § result
0ϩ2
12 ϩ 1Ϫ0.22
2
11.8
Now try Exercises 25 through 28
ᮣ
Matrix Multiplication
Consider a cable company offering three different levels of Internet service: Bronze—
fast, Silver—very fast, and Gold—lightning fast. Table 7.1 shows the number and
types of programs sold to households and businesses for the week. Each program has
an incentive package consisting of a rebate and a certain number of free weeks, as
shown in Table 7.2.
Table 7.1 Matrix A
Bronze
Silver
Table 7.2 Matrix B
Gold
Rebate
Free Weeks
Homes
40
20
25
Bronze
$15
2
Businesses
10
15
45
Silver
$25
4
Gold
$35
6
To compute the amount of rebate money the cable company paid to households
for the week, we would take the first row (R1) in Table 7.1 and multiply by the
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655
corresponding entries (bronze with bronze, silver with silver, and so on) in the first
column (C1) of Table 7.2 and add these products. In matrix form, we have
15
#
3 40 20 25 4 £ 25 § ϭ 40 # 15 ϩ 20 # 25 ϩ 25 # 35 ϭ $1975. Using R1 of Table 7.1
35
with C2 from Table 7.2 gives the number of free weeks awarded to homes:
2
#
3 40 20 254 £ 4 § ϭ 40 # 2 ϩ 20 # 4 ϩ 25 # 6 ϭ 310. Using the second row (R2) of
6
Table 7.1 with the two columns from Table 7.2 will give the amount of rebate money
and the number of free weeks, respectively, awarded to business customers. When all
computations are complete, the result is a product matrix P with order 2 ϫ 2. This is
because the product of R1 from matrix A, with C1 from matrix B, gives the entry in
15 2
40 20 25 #
1975 310
d £ 25 4 § ϭ c
d.
position P11 of the product matrix: c
10 15 45
2100 350
35 6
#
Likewise, the product R1 C2 will give entry P12 (310), the product of R2 with C1 will
give P21 (2100), and so on. This “row ϫ column” multiplication can be generalized,
and leads to the following. Given m ϫ n matrix A and s ϫ t matrix B,
A
1m ϫ n2
c
B
1s ϫ t 2
A
1m ϫ n2
c
c
matrix multiplication is
possible only when
nϭs
B
1s ϫ t 2
c
result will be an
m ϫ t matrix
In more formal terms, we have the following definition of matrix multiplication.
Matrix Multiplication
Given the m ϫ n matrix A ϭ 3aij 4 and the s ϫ t matrix B ϭ 3bij 4. If n ϭ s, then
matrix multiplication is possible and the product AB is an m ϫ t matrix P ϭ 3 pij 4,
where pij is product of the ith row of A with the jth column of B.
In less formal terms, matrix multiplication involves multiplying the row entries of
the first matrix with the corresponding column entries of the second, and adding them
together. In Example 6, two of the matrix products [parts (a) and (b)] are shown in full
detail, with the first entry of the product matrix color-coded.
EXAMPLE 6
ᮣ
Multiplying Matrices
Given the matrices A through E shown here, compute the following products:
a. AB
Ϫ2
Aϭ c
3
b. CD
1
d
4
4
Bϭ c
6
3
d
1
c. DC
Ϫ2 1
Cϭ £ 1
0
4
1
d. AE
3
2 §
Ϫ1
2
D ϭ £4
0
e. EA
5
1
Ϫ1
1 §
3
Ϫ2
Ϫ2
Eϭ £ 3
1
Ϫ1
0 §
2
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CHAPTER 7 Matrices and Matrix Applications
Solution
ᮣ
Ϫ2
1 4 3
d ϭ c
dc
36
4 6 1
1Ϫ22142 ϩ 112162
Computation: c
132142 ϩ 142 162
a. AB ϭ c
Ϫ2
3
A
B
(2 ϫ 2) (2 ϫ 2)
Ϫ5
d
13
1Ϫ22 132 ϩ 112 112
d
132132 ϩ 142112
c
0 Ϫ2 Ϫ7
Ϫ2 1 3
2 5
1
b. CD ϭ £ 1 0 2 § £ 4 Ϫ1 1 § ϭ £ 2 11 Ϫ3 §
12 16 7
4 1 Ϫ1 0 3 Ϫ2
1Ϫ22122 ϩ 112142 ϩ (3)(0)
Computation: £ 112122 ϩ 102 142 ϩ 122(0)
142 122 ϩ 112 142 ϩ 1Ϫ12(0)
Ϫ2
1
d£ 3
4
1
Ϫ2
e. EA ϭ £ 3
1
Ϫ1
Ϫ2
0 §c
3
2
c
C
D
(3 ϫ 3) (3 ϫ 3)
C
D
(3 ϫ 3) (3 ϫ 3)
c
c
c
c
result will be a
3 ϫ 3 matrix
1Ϫ22 112 ϩ 112112 ϩ (3)1Ϫ22
112112 ϩ 102112 ϩ 1221Ϫ22 §
142112 ϩ 112112 ϩ 1Ϫ121Ϫ22
D
(3 ϫ 3)
c
C
(3 ϫ 3)
D
(3 ϫ 3)
c
c
multiplication is possible
since 3 ϭ 3
A
(2 ϫ 2)
Ϫ1
0 §
2
c
result will be a
2 ϫ 2 matrix
1Ϫ22152 ϩ 112 1Ϫ12 ϩ (3)132
112152 ϩ 1021Ϫ12 ϩ 122132
142 152 ϩ 1121Ϫ12 ϩ 1Ϫ12132
1
1
d ϭ £ Ϫ6
4
4
c
multiplication is possible
since 2 ϭ 2
multiplication is possible
since 3 ϭ 3
2 5
1
Ϫ2 1 3
5
3 15
c. DC ϭ £ 4 Ϫ1 1 § £ 1 0 2 § ϭ £Ϫ5 5 9 §
0 3 Ϫ2
4 1 Ϫ1
Ϫ5 Ϫ2 8
Ϫ2
d. AE ϭ c
3
A
B
(2 ϫ 2) (2 ϫ 2)
c
C
(3 ϫ 3)
c
result will be a
3 ϫ 3 matrix
E
(3 ϫ 2)
c
multiplication is not possible since 2
E
(3 ϫ 2)
Ϫ6
3 §
9
c
A
(2 ϫ 2)
E
(3 ϫ 2)
c
3
A
(2 ϫ 2)
c
multiplication is possible
since 2 ϭ 2
c
result will be a
3 ϫ 2 matrix
Now try Exercises 29 through 40
ᮣ
Example 6 shows that in general, matrix multiplication is not commutative. Parts
(b) and (c) show CD DC since we get different results, and parts (d) and (e) show
AE EA, since AE is not defined while EA is.
As with the addition and subtraction of matrices, matrix multiplication becomes
cumbersome and time consuming for larger matrices, and we will often turn to the
technology available in such cases.
EXAMPLE 7
ᮣ
Using Technology for Matrix Operations
Use a calculator to compute the product AB.
Solution
ᮣ
2
Ϫ1
Aϭ ≥
6
3
Carefully enter matrices A and B into the calculator,
then press 2nd MODE (QUIT) to get to the home
screen. Use [A][B] , and the calculator finds the
product shown in the figure. Just for “fun,” we’ll
Ϫ3
5
0
2
0
1
2
4
¥ B ϭ £ 0.5
2
Ϫ2
Ϫ1
A
(4 ϫ 3)
Ϫ0.7
3.2
3
4
B
(3 ϫ 3)
A
(4 ϫ 3)
c
c
1
Ϫ3 §
4
B
(3 ϫ 3)
ENTER
c
multiplication is possible
since 3 ϭ 3
c
result will be a
4 ϫ 3 matrix
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Section 7.2 The Algebra of Matrices
double check the first entry of the product matrix:
(2)(1/2) ϩ (Ϫ3)(0.5) ϩ (0)(Ϫ2) 0.5.
2
1
AB
6
3
3
5
0
2
0
1
2
4
Ơ Ê 0.5
2
2
1
0.7
3.2
3
4
1
3 Đ
4
Now try Exercises 41 through 52
ᮣ
Properties of Matrix Multiplication
Earlier, Example 6 demonstrated that matrix multiplication is not commutative. Here
is a group of properties that do hold for matrices. You are asked to check these properties in the Exercise Set using various matrices. See Exercises 53 through 56.
Properties of Matrix Multiplication
Given matrices A, B, and C for which the products are defined:
I. A1BC2 ϭ 1AB2C
II. A1B ϩ C2 ϭ AB ϩ AC
III. 1B ϩ C2A ϭ BA ϩ CA
IV. k1A ϩ B2 ϭ kA ϩ kB
S matrix multiplication is associative
S matrix multiplication is distributive from the left
S matrix multiplication is distributive from the right
S a constant k can be distributed over addition
We close this section with an application of matrix multiplication. There are many
other interesting applications in the Exercise Set.
EXAMPLE 8
ᮣ
Using Matrix Multiplication to Track Volunteer Enlistments
In a certain country, the number of males and females that will join the military depends on their age.
This information is stored in matrix A (Table 7.3). The likelihood a volunteer will join a particular
branch of the military also depends on their age, with this information stored in matrix B (Table 7.4).
(a) Compute the product P ϭ AB and discuss/interpret what is indicated by the entries P11, P13, and
P24 of the product matrix. (b) How many males are expected to join the Navy this year?
Table 7.4 Matrix B
Table 7.3 Matrix A
A
Solution
ᮣ
B
Age Group
Likelihood of Joining
Sex
18–19
20–21
22–23
Age Group
Army
Navy
Air Force
Marines
Female
1000
1500
500
18–19
0.42
0.28
0.17
0.13
Male
2500
3000
2000
20–21
0.38
0.26
0.27
0.09
22–23
0.33
0.25
0.35
0.07
a. Matrix A has order 2 ϫ 3 and matrix B has order 3 ϫ 4. The product matrix P can be found
and is a 2 ϫ 4 matrix. Carefully enter the matrices in your calculator. Figure 7.12 shows the
entries of matrix B. Using 3A 4 3 B4 , the calculator finds the product matrix shown in
Figure 7.13. Pressing the right arrow shows the complete product matrix is
ENTER
Pϭ c
1155
2850
795
1980
750
1935
300
d.
735
The entry P11 is the product of R1 from A and C1 from B, and indicates that for the year, 1155
females are projected to join the Army. In like manner, entry P13 shows that 750 females are
projected to join the Air Force. Entry P24 indicates that 735 males are projected to join the
Marines.
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CHAPTER 7 Matrices and Matrix Applications
Figure 7.13
Figure 7.12
b. The product R2 (males) # C2 (Navy) gives P22 ϭ 1980, meaning 1980 males are expected to
join the Navy.
Now try Exercise 59 through 66
C. You’ve just seen how
we can compute the product
of two matrices
ᮣ
7.2 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
ᮣ
1. Two matrices are equal if they are like size and the
corresponding entries are equal. In symbols, A ϭ B
if
.
ϭ
2. The sum of two matrices (of like size) is found by
adding the corresponding entries. In symbols,
AϩBϭ
.
3. The product of a constant times a matrix is called
multiplication.
4. The size of a matrix is also referred to as its
1 2 3
d is
The order of A ϭ c
.
4 5 6
5. Give two reasons why matrix multiplication is
generally not commutative. Include several
examples using matrices of various sizes.
6. Discuss the conditions under which matrix
multiplication is defined. Include several examples
using matrices of various sizes.
.
DEVELOPING YOUR SKILLS
State the order of each matrix and name the entries in
positions a12 and a23 if they exist. Then name the
position aij of the “5” in each.
1
7. c
5
2
9. c
0
Ϫ2
11. £ 0
5
Ϫ3
5
1
8
Ϫ1
13. c
19
8. £ Ϫ11 §
5
Ϫ3
d
Ϫ7
0.5
d
6
Ϫ7
1§
4
2
10. £ Ϫ0.1
0.3
89
12. £ 13
2
Determine if the following statements are true, false, or
conditional. If false, explain why. If conditional, find values
of a, b, c, p, q, and r that will make the statement true.
3
2
14. ≥
Ϫ1
2
0.4
5§
Ϫ3
55
8
1
34
5
1
21
3§
0
14
132
11
116
Ϫ2
15. £ 2b
0
Ϫ7
5
Ϫ2
5
3
Ϫ5
Ϫ9
2p ϩ 1
16. Ê 1
q5
18
1
d c
164
4
2
4 12
13
10
1.5
Ơ c
1
0.5
3
1.4
0.4
a
c
4 Đ £6
3c
0
Ϫ5
12
9
3
Ϫ5
Ϫ3b
9
7
0 § ϭ £ 1
Ϫ2r
Ϫ2
2 12
d
8
1.3
d
0.3
Ϫ4
Ϫa §
Ϫ6
Ϫ5
3r
3p
2Ϫq
0 §
Ϫ8
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For matrices A through J as given, perform the
indicated operation(s), if possible. If an operation
cannot be completed, state why. Use a calculator only
for those exercises designated by an icon.
Aϭ c
2
5
1
E ϭ £0
4
0.5
5 §
3
Ϫ2
Ϫ1
3
Ϫ1
Gϭ £ 0
Ϫ4
1
2
1
4
1
8
2
Bϭ £ 1 §
Ϫ3
3
d
8
2
C ϭ Ê 0.2
1
I
0
2 Đ
6
2
1
3
3
8
3
2
3
4
1
D Ê0
0
0
2 Đ
6
1
4
5
8
5
2
F c
H c
0
1
0
6
12
8
5
9
d
6
3
d
2
7
32
J
5
16
A c
5
3
5
8
Ơ
3
16
0
G
1
2
1
4
46. BH
47. DG
48. GD
2
22. A Ϫ J
23. G ϩ I
24. I Ϫ G
25. 3H Ϫ 2A
26. 2E ϩ 3G
32. HA
33. CB
34. FH
35. HF
36. EB
37. H
2
39. FE
38. F2
40. EF
1
4
Ϫ3
1
19
Hϭ ≥
8
1
1
19
16
3
4
3
8
11
16
45. HB
21. H ϩ J
0
0§
1
0
12 Ϫ8 32
2 § Fϭ c
d
4
8 16
Ϫ6
44. GE
51. FG
31. AH
0
1
0
43. EG
20. G ϩ D
30. DE
1
D ϭ £0
0
0
d
1
42. HA
19. F ϩ H
29. ED
1
0
41. AH
49. C
2
28. F Ϫ F
3
Bϭ c
13
3 §
2 13
Ϫ2
Ϫ1
3
1
E ϭ £0
4
18. E ϩ G
1
E Ϫ 3D
2
4
d
9
13
Cϭ £ 2
13
0
0§
1
Ϫ3
0
For matrices A through H as given, use a calculator to
perform the indicated operation(s), if possible. If an
operation cannot be completed, state why.
17. A ϩ H
27.
659
Section 7.2 The Algebra of Matrices
4
57
¥
5
57
50. E2
52. AF
For Exercises 53 through 56, use a calculator and
matrices A, B, and C to verify each statement.
Ϫ1
Aϭ £ 2
4
3
7
0
45
C ϭ £ Ϫ6
21
Ϫ1
10
Ϫ28
5
Ϫ1 §
6
0.3
B ϭ £ Ϫ2.5
1
Ϫ0.4
2
Ϫ0.5
1.2
0.9 §
0.2
3
Ϫ15 §
36
53. Matrix multiplication is not generally commutative:
(a) AB BA, (b) AC CA, and (c) BC CB.
54. Matrix multiplication is distributive from the left:
A1B ϩ C2 ϭ AB ϩ AC.
55. Matrix multiplication is distributive from the right:
1B ϩ C2A ϭ BA ϩ CA.
56. Matrix multiplication is associative:
1AB2C ϭ A1BC2.
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ᮣ
c
7–24
CHAPTER 7 Matrices and Matrix Applications
WORKING WITH FORMULAS
2
W
2
d
0
#
c
L
Perimeter
d ؍c
d
W
Area
The perimeter and area of a rectangle can be simultaneously calculated using the matrix formula shown, where L
represents the length and W represents the width of the rectangle. Use the matrix formula and your calculator to find
the perimeter and area of the rectangles shown, then check the results using P ϭ 2L ϩ 2W and A ϭ LW.
57.
6.374 cm
4.35 cm
ᮣ
58.
5.02 cm
3.75 cm
APPLICATIONS
59. Custom T’s designs and sells specialty T-shirts and
sweatshirts, with factories in Verdi and Minsk. The
company offers this apparel in three quality levels:
standard, deluxe, and premium. Last fall the Verdi
plant produced 3820 standard, 2460 deluxe, and
1540 premium T-shirts, along with 1960 standard,
1240 deluxe, and 920 premium sweatshirts. The
Minsk plant produced 4220 standard, 2960 deluxe,
and 1640 premium T-shirts, along with 2960
standard, 3240 deluxe, and 820 premium
sweatshirts in the same time period.
a. Write a 3 ϫ 2 “production matrix” for each
plant 3 V S Verdi, M S Minsk], with a T-shirt
column, a sweatshirt column, and three rows
showing how many of the different types of
apparel were manufactured.
b. Use the matrices from part (a) to determine
how many more or fewer articles of clothing
were produced by Minsk than Verdi.
c. Use scalar multiplication to find how many
shirts of each type will be made at Verdi and
Minsk next fall, if each is expecting a 4%
increase in business.
d. Write a matrix that shows Custom T’s total
production next fall (from both plants), for
each type of apparel.
60. Terry’s Tire Store sells automobile and truck tires
through three retail outlets. Sales at the Cahokia store
for the months of January, February, and March
broke down as follows: 350, 420, and 530 auto tires
and 220, 180, and 140 truck tires. The Shady Oak
branch sold 430, 560, and 690 auto tires and 280,
320, and 220 truck tires during the same 3 months.
Sales figures for the downtown store were 864, 980,
and 1236 auto tires and 535, 542, and 332 truck tires.
a. Write a 2 ϫ 3 “sales matrix” for each store
3 C S Cahokia, S S Shady Oak, D S
Downtown], with January, February, and
March columns, and two rows showing the
sales of auto and truck tires respectively.
b. Use the matrices from part (a) to determine
how many more or fewer tires of each type the
downtown store sold (each month) over the
other two stores combined.
c. Market trends indicate that for the same three
months in the following year, the Cahokia
store will likely experience a 10% increase in
sales, the Shady Oak store a 3% decrease, with
sales at the downtown store remaining level
(no change). Write a matrix that shows the
combined monthly sales from all three stores
next year, for each type of tire.
61. Home improvements: Dream-Makers Home
Improvements specializes in replacement windows,
replacement doors, and new siding. During the
peak season, the number of contracts that came
from various parts of the city (North, South, East,
and West) are shown in matrix C. The average
profit per contract is shown in matrix P. Compute
the product PC and discuss what each entry of the
product matrix represents.
Windows
Doors
Siding
Windows
3 1500
N S E W
9 6 5 4
£7 5 7 6§ ϭ C
2 3 5 2
Doors Siding
500
2500 4 ϭ P
62. Classical music: Station 90.7 — The Home of
Classical Music — is having their annual fund
drive. Being a loyal listener, Mitchell decides that
for the next 3 days he will donate money according
to his favorite composers, by the number of times
their music comes on the air: $3 for every piece by
Mozart (M), $2.50 for every piece by Beethoven
(B), and $2 for every piece by Vivaldi (V).
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College Algebra G&M—
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Section 7.2 The Algebra of Matrices
This information is displayed in matrix D. The
number of pieces he heard from each composer is
displayed in matrix C. Compute the product DC
and discuss what each entry of the product matrix
represents.
Mon. Tue. Wed.
M 4
3
5
B £3
2
4§ ϭ C
V 2
3
3
M B V
3 3 2.5 2 4 ϭ D
63. Pizza and salad: The science department and math
department of a local college are at a pre-semester
retreat, and decide to have pizza, salads, and soft
drinks for lunch. The quantity of food ordered by
each department is shown in matrix Q. The cost of
the food item at each restaurant is shown in matrix
C using the published prices from three popular
restaurants: Pizza Home (PH), Papa Jeff’s (PJ), and
Dynamos (D).
a. What is the total cost to the math department if
the food is ordered from Pizza Home?
b. What is the total cost to the science department
if the food is ordered from Papa Jeff’s?
c. Compute the product QC and discuss the
meaning of each entry in the product matrix.
Pizza
Science 8
c
Math 10
PH
Pizza
8
Salad £ 1.5
Drink 0.90
Salad Drink
12
20
d ϭQ
8
18
PJ
D
7.5
10
1.75
2 § ϭC
1
0.75
64. Manufacturing pool tables: Cue Ball
Incorporated makes three types of pool tables, for
homes, commercial use, and professional use. The
amount of time required to pack, load, and install
each is summarized in matrix T, with all times in
hours. The cost of these components in dollars per
hour, is summarized in matrix C for two of its
warehouses, one on the west coast and the other in
the midwest.
a. What is the cost to package, load, and install a
commercial pool table from the coastal
warehouse?
b. What is the cost to package, load, and install a
commercial pool table from the warehouse in
the midwest?
c. Compute the product TC and discuss the
meaning of each entry in the product matrix.
661
Pack Load Install
Home
1
0.2 1.5
Comm £ 1.5
0.5 2.2 § ϭ T
Prof 1.75 0.75 2.5
Coast Midwest
Pack 10
8
Load £ 12
10.5 § ϭ C
Install 13.5 12.5
65. Joining a club: Each school year, among the
students planning to join a club, the likelihood a
student joins a particular club depends on their
class standing. This information is stored in matrix
C. The number of males and females from each
class that are projected to join a club each year is
stored in matrix J. Compute the product JC and use
the result to answer the following:
a. Approximately how many females joined the
chess club?
b. Approximately how many males joined the
writing club?
c. What does the entry p13 of the product matrix
tells us?
Fresh
Female 25
c
Male 22
Spanish
Fresh 0.6
Soph £ 0.5
Junior 0.4
Soph Junior
18
21
d ϭJ
19
18
Chess
0.1
0.2
0.2
Writing
0.3
0.3 § ϭ C
0.4
66. Designer shirts: The SweatShirt Shoppe sells
three types of designs on its products: stenciled (S),
embossed (E), and applique (A). The quantity of
each size sold is shown in matrix Q. The retail
price of each sweatshirt depends on its size and
whether it was finished by hand or machine. Retail
prices are shown in matrix C. Assuming all stock is
sold, compute the product QC and use the result to
answer the following.
a. How much revenue was generated by the large
sweatshirts?
b. How much revenue was generated by the
extra-large sweatshirts?
c. What does the entry p11 of the product matrix
QC tell us?
S
med 30
large £ 60
x-large 50
E
30
50
40
A
15
20 § ϭ Q
30
Hand
S 40
E £ 60
A 90
Machine
25
40 § ϭ C
60
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ᮣ
EXTENDING THE CONCEPT
67. For the matrix A shown, use your calculator to
compute A2, A3, A4, and A5. Do you notice a
pattern? Try to write a “matrix formula” for An,
where n is a positive integer, then use your formula
to find A6. Check results using a calculator.
1 0 1
A ϭ £1 1 1§
1 0 1
ᮣ
7–26
CHAPTER 7 Matrices and Matrix Applications
68. The matrix M ϭ c
2
1
d has some very
Ϫ3 Ϫ2
interesting properties. Compute the powers M 2, M 3,
M 4, and M 5, then discuss what you find. Try to
find/create another 2 ϫ 2 matrix that has similar
properties.
MAINTAINING YOUR SKILLS
69. (6.2) Solve the system using elimination.
x ϩ 2y Ϫ z ϭ 3
• Ϫ2x Ϫ y ϩ 3z ϭ Ϫ5
5x ϩ 3y Ϫ 2z ϭ 2
log2 21
70. (2.1/4.6) Given f(x) ϭ x ϩ 10x Ϫ 9, solve
f 1x2 Ն 0.
4
ᮣ
2
MID-CHAPTER
CHECK
MID-CHAPTER
Ϫ2
2. B ϭ c
4
1.1
0.1
0.4
1
1
2
3
4
0
0.2
Ϫ0.9 §
0.8
5
d
Ϫ3
Write each system in matrix form and solve using row
operations to triangularize the matrix. If the system is
linearly dependent, write the solution using a parameter.
3. e
Ϫx ϩ y Ϫ 5z ϭ 23
4. • 2x ϩ 4y Ϫ z ϭ 9
3x Ϫ 5y ϩ z ϭ 1
2x ϩ 3y ϭ Ϫ5
Ϫ5x Ϫ 4y ϭ 2
x ϩ y Ϫ 3z ϭ Ϫ11
5. • 4x Ϫ y Ϫ 2z ϭ Ϫ4
3x Ϫ 2y ϩ z ϭ 7
6. For matrices A and B given, compute:
Aϭ c
a. A Ϫ B
Ϫ3
5
Ϫ2
10
d Bϭ c
4
Ϫ30
b.
2
B
5
72. (4.1) Find the quotient using synthetic division,
then check using multiplication.
x3 Ϫ 9x ϩ 10
xϪ2
CHECK
State the size of each matrix and identify the entry in
second row, third column.
0.4
1. A ϭ £ Ϫ0.2
0.7
71. (5.4) Evaluate using the change-of-base formula,
then check using exponentiation.
7. For matrices C and D given, use a calculator to find:
Ϫ0.2
C ϭ £ 0.4
0.1
c. 5A ϩ B
0.2
5
0 § D ϭ £ Ϫ2.5
Ϫ0.1
10
2.5
0
2.5
10
Ϫ5 §
10
1
a. C ϩ D
b. Ϫ0.6D
c. CD
5
8. For the matrices A, B, C, and D given, compute the
products indicated (if possible):
4
Aϭ c
0
4
Cϭ c
Ϫ1
a. AC
15
d
Ϫ5
0
0.8
Ϫ0.2
Ϫ8
0
Ϫ2
1 §
7
6
B ϭ £0
4
Ϫ1
d
Ϫ5
Ϫ3
d
1
b. Ϫ2CD
2
D ϭ £ Ϫ1
1
c. BA
0
Ϫ3
5
Ϫ6
0 §
Ϫ4
d. CBϪ 4A
9. Create a system of equations to model this exercise,
then write the system in matrix form and solve. The
campus bookstore offers both new and used texts to
students. In a recent biology class with 24 students,
14 bought used texts and 10 bought new texts, with
the class as a whole paying $2370. Of the 6 premed
students in class, 2 bought used texts, and 4 bought
new texts, with the group paying a total of $660.
How much does a used text cost? How much does a
new text cost?
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Section 7.3 Solving Linear Systems Using Matrix Equations
10. Table A shown gives the number and type of extended warranties sold to individual car owners and to business
fleets. Table B shows the promotions offered to those making the purchase. Write the entries of each table in
matrix form and compute the product matrix P ϭ AB, then state what each entry of the product matrix represents.
Table B
Table A
Rebate
Free AAA
Membership
$50
1 yr
100,000 mi
$75
2 yr
120,000 mi
$100
3 yr
Extended
Warranties
80,000
mi
100,000
mi
120,000
mi
Individuals
30
25
10
80,000 mi
Businesses
20
12
5
Promotions
REINFORCING BASIC CONCEPTS
More on Matrix Multiplication
To help understand and master the concept of matrix multiplication, it helps to take a closer
look at the entries of the product matrix. Recall for the product AB ϭ P, the entry p11 in the
product matrix is the result of multiplying the 1st row of A with the 1st column of B, the entry
p12 is the result of multiplying 1st row of A, with the 2nd column of B, and so on.
1
P ϭ £6
7
2
Ϫ2 §
7
Exercise 1: The product of the 3rd row of A with the 2nd column of B, gives what entry in P?
Exercise 2: The entry p13 is the result of what product? The entry p22 is the result of what product?
Exercise 3: If p33 is the last entry of the product matrix, what are the possible sizes of A and B?
Exercise 4: Of the eight matrices shown here, only two produce the product matrix P shown. Use the ideas
highlighted above to determine which two.
3
A ϭ £2
4
Ϫ1
1 §
1
1
Bϭ c
2
1
d
0
2
C ϭ £ Ϫ1
1
1
Eϭ c
3
2
d
Ϫ1
1
F ϭ £0
4
0
2§
1
1
G ϭ £4§
6
7.3
0
Ϫ3
5
Ϫ6
0 §
Ϫ4
2
D ϭ £ Ϫ1
3
H ϭ 31
4
0§
5
34
Solving Linear Systems Using Matrix Equations
LEARNING OBJECTIVES
In Section 7.3 you will see
how we can:
A. Recognize the identity
matrix for multiplication
B. Find the inverse of a
square matrix
C. Solve systems using
matrix equations
D. Use determinants to find
whether a matrix is
invertible
While using matrices and row operations offers a degree of efficiency in solving systems, we are still required to solve for each variable individually. Using matrix multiplication we can actually rewrite a given system as a single matrix equation, in which the
solutions are computed simultaneously. As with other kinds of equations, the use of
identities and inverses are involved, which we now develop in the context of matrices.
A. Multiplication and Identity Matrices
From the properties of real numbers, 1 is the identity for multiplication since
n # 1 ϭ 1 # n ϭ n. A similar identity exists for matrix multiplication. Consider the 2 ϫ 2
1 4
matrix A ϭ c
d . While matrix multiplication is not generally commutative,
Ϫ2 3